First, a centralizer \(\mathcal {C}_{\mathcal {D}}(L)\)  is known to be a finitely generated free \({\bf C}[L]\) -module and we use a result by Goodearl in [1]} to the effect that the cardinality of any basis is a divisor of \(n=\operatorname{ord}(L)\) . By restricting attention to polynomial coefficients, in Section  we determine the initial form of the elements in the centralizer of \(L\) , by automating the “Dixmier test” by means of a suitable filtration. As a consequence, we can guarantee in Section REF  that the centralizer of an operator of order 4 in the first Weyl algebra \(A_{1} ({\bf C})\)  is the ring of a plane algebraic curve in \({\bf C}^2\)  (this, given that all centralizers are affine rings of irreducible, though not necessarily reduced, curves, amounts to saying that there is a plane model of the curve which only misses one
smooth point at infinity, cf. [2]}, or equivalently, that the centralizer can be generated by two elements).
